Tagged Questions
1
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0answers
25 views
Uniformly quasi-isometric patches over classes of Riemannian manifolds
Suppose $(M^d,g)$ is a closed, connected Riemannian manifold.
Is there a constant $R > 0$ such that for all $z \in (M,g)$, for all $x \in B_R(z)$, \begin{equation} \frac{1}{2} \lVert \xi ...
1
vote
1answer
31 views
Space of embedded surfaces with a common point
Consider the space of all embedded orientable surfaces in $ R^3 $ of constant mean curvature (the minimal case is included) passing through the origin. I'm asking if there exists a topology on this ...
5
votes
0answers
45 views
What is the volume of Complex Projective Space with Fubini-Study Metric?
I try to compute the volume of the complex projective space $\mathbb{CP}^n$ with Fubini-Study metric, normalized to have diameter $=\pi/2$ i.e. the sectional curvatures lie between $1$ and $4$. Fix a ...
2
votes
1answer
44 views
Prove Green formula
Let $(M^n,g)$ be an oriented Riemannian manifold with boundary $\partial M$.
The orientation on $Μ$ defines an orientation on $\partial M$. Locally, on the boundary, choose a positively oriented ...
2
votes
1answer
45 views
How to conclude $\frac{d}{dt}E(f)=-\int _M (\Delta f)\dot{f} d\mu$?
Can anyone explain to me how I can conclude $\frac{d}{dt}E(f)=-\int _M (\Delta f)\dot{f} d\mu$ by using integration by parts and $\langle f_1 ,f_2 \rangle_\mu:=\int_M f_1 f_2 d\mu$?
Where $M$ is a ...
2
votes
0answers
63 views
Exponential Map
this seems to be an easy question but I'm stuck anyway. Let $\Gamma$ be a submanifold of a Riemannian manifold $(M,g)$. Let further $U$ be a coodinate neighborhood
of $\Gamma$ such that a point ...
1
vote
0answers
40 views
Is there a relation between Super Riemannian manifolds and Kahler manifolds?
(This question has a physics motivation).
Could we establish any kind of relationship between Super Riemannian manifolds (super Riemannian structures on super manifolds) and Kahler manifolds, or at ...
2
votes
0answers
50 views
Geodesics and Christoffel symbols
If these are satisfied then we are on a geodesic. Do I just need to plug in the condition given about the christoffel symbols and then see that the equations are allways fulfiled as long as $v=at+b$ ...
1
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1answer
40 views
Orientability of $P_{\bf R}T{\bf RP}^{2n}$
I know the following fact :
(1) $ {\bf RP}^{2n}$ is non-orientable.
(2) $ {\bf RP}^{2n-1}$ is orientable.
(3) $P_{\bf R}T{\bf RP}^{2n}$ is orientable.
(4) $P_{\bf R}T{\bf RP}^{2n+1}$ ...
2
votes
1answer
73 views
Gradient in Riemannian manifold
I have a calculation involving a gradient and a parametrization, but I haven't been able to find out the relation between them. Let me explain.
Let $f:X↦R$ be a smooth function and $\mathrm{grad}f\in ...
2
votes
1answer
29 views
what is the inner product appeared in front of the integral?
Given a compact manifold with a Riemannian metric $g$, we define the total
scalar curvature by
$$E(g)=\int_M RdV$$
Let us consider the first variation of $E$ under an arbitrary change of metric. We ...
1
vote
1answer
32 views
Meaning of modulo diffeomorphism
I faced this sentence:
we consider the space of Riemannian metrics modulo diffeomorphism and scaling.
Can anyone explain to me what is the meaning of modulo diffeomorphism and scaling?
Thanks!
5
votes
2answers
88 views
Hopf's theorem on CMC surfaces
I got stuck reading the proof of the following theorem:
Theorem (Heinz Hopf) Let $X: S^2\to \mathbb R^3$ be a constant mean curvature immersion. Then $X(S^2)$ is a round sphere.
Proof: Let ...
2
votes
0answers
69 views
Lie bracket of vector fields definition equivalence
Lie bracket of vector fields is defined in two ways:
Let $\Phi^X_t$ be the flow associated with the vector field $X$, and let $d$ denote the
tangent map derivative operator. Then
the Lie ...
1
vote
0answers
73 views
Energy functional
During my study on Ricci Flow I faced some functional known as enery functional. For example Einstein-Hilbert functional is called an energy functional, also in Perelman's works ...
2
votes
1answer
77 views
Gradient of a functional
Given a compact manifold with a Riemannian metric $g$, we define the total
scalar curvature by
$$E(g)=\int_M RdV$$
Let us consider the first variation of $E$ under an arbitrary change of metric. We ...
5
votes
2answers
126 views
Justification for this manipulation in a proof of the first variation of energy formula
As a part of my current homework assignment, I am to derive the first variation of energy identity. Working out the problem with my friends, we came to exactly the same argument as presented in these ...
1
vote
1answer
72 views
Quotient theorem for tensors
Can somebody please explain to me how the following statement is true?
The Riemann curvature tensor $R^c_{dab}$ is given by the Ricci identity $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv ...
4
votes
0answers
40 views
Metric on Steifel and Grassmannian manifolds generalizing Fubini-Study
If $F$ is $\mathbb{R}, \mathbb{C}$, or $ \mathbb{H}$, the Grassmannian manifold $G_k(\textbf F^n)$ is the space of all $k$ dimensional subspaces of the $n$ dimensional vector space $F^n$. The Stiefel ...
1
vote
1answer
40 views
Confusion regarding uniqueness of Levi-Civita connection
Assuming a Levi-Civita connection exists it is uniquely determined.
Using $\nabla g = 0$ and the symmetry of the metric tensor $g$ we
find:
$ X (g(Y,Z)) + Y (g(Z,X)) - Z (g(Y,X)) = ...
4
votes
2answers
109 views
geometric interpretation of Lie bracket
On page 159 of "A Comprehensive Introduction To Differential Geometry Vol.1" by Spivak has written:
We thus see that the bracket $[X,Y]$ measures, in some sense, the extent to
which the integral ...
1
vote
0answers
63 views
Riemannian curvature and its application on covariant derivative of tensors
This identity can be generalized to get the commutators for two
covariant derivatives of arbitrary tensors as follows $ \begin{align} T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s ; ...
0
votes
0answers
44 views
Existence of Solution: Embedding from 2D Euclidean space to a circle
Given a real matrix $X$ with $n$ rows and 2 columns, can the matrix be transformed to a real matrix $Y$ such that all the points formed by the rows of $Y$ lie on a circle (2d) and their inter-point ...
0
votes
1answer
31 views
Question regarding Nash-Kuiper embedding theorem
In Wikipedia description of Nash-Kuiper theorem, it says:
Let $(M,g)$ be a Riemannian manifold and $f: M^{m} \rightarrow \mathbb{R}^n$ a short $C^{\infty}$-embedding into Euclidean space
...
1
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0answers
43 views
Reason for defining Riemannian curvature tensor and torsion tensor in particular way
I saw how Riemannian curvature tensor and torsion tensor are defined, but I am not sure why they are defined that way. In 3-dimensional euclidean space with ordinary multivariable calculus, the ...
4
votes
1answer
86 views
Problem about Ricci Flow
On page 12 of "Lectures On Ricci Flow" by Peter Topping is written:
In two dimensions, we know that the Ricci curvature can be written in
terms of the Gauss curvature $K$ as $Ric(g) = Kg$. Working ...
4
votes
1answer
53 views
covariant derivative vs. exterior derivative
I have the following question. Let $M$ be a Riemannian manifold with metric $g$ and $\nabla$ the Levi-Civita connection. Let furthermore $\alpha \in \Omega^{k}(M)$ be a $k$-form such that $\nabla ...
2
votes
1answer
58 views
the Ricci curvature in two dimension
In two dimensions, we know that the Ricci curvature can be written in
terms of the Gauss curvature K as $Ric(g) = Kg$.
Can anyone prove this?
Sorry if the question is too trivial :).
0
votes
1answer
72 views
About Sectional Curvature
In a paper by Yann Ollivier:
Let $x$ be a point in $X$, $v$ a small tangent vector at $x$, $y$ the endpoint
of $v$, $w_x$ a small tangent vector at $x$, and $w_y$ the parallel transport of $w_x$ from ...
1
vote
2answers
50 views
Showing that the riemanian metric $\frac{g}{\sqrt{x^2+y^2+z^2}}$ is complete
I would like to show that a certain Riemannian metric defined on $\mathbb{R^3}$ is complete. The metric is given in the following sentence from this article (pg 160):
... a Riemannian metric on ...
1
vote
1answer
36 views
Elementary definition: what's a parallel volume-form?
This is a very elementary question,
What is the definition for a volume form (or $n$-form) to be parallel with respect to the metric?
To find out more about the concept, what kind of topic do I need ...
1
vote
1answer
95 views
Curvature of a metric defined on an open disc in $\mathbb{R}^2$
Let $D$ be an open disc centred at the origin in $\mathbb{R}^2$. Give $D$ a Riemannian metric of the the form $(dx^2+dy^2)/f(r)^2$, where $r=\sqrt{x^2+y^2}$ and $f(r)>0$. Show that the curvature of ...
1
vote
1answer
81 views
Difference between “Live” and “Define”
In many mathematical text to determine an object on manifold, the verbs "live" and "define" are used.
I'm interested to know whether there is a difference between the concepts of "to define" and "to ...
1
vote
1answer
63 views
Parallel transport for a conformally equivalent metric
Suppose $M$ is a smooth manifold equipped with a Riemannian metric $g$. Given a curve $c$, let $P_c$ denote parallel transport along $c$. Now suppose you consider a new metric $g'=fg$ where $f$ is a ...
1
vote
0answers
35 views
A $k$-form is thought of as measuring the flux through an infinitesimal k-parallelepiped
On the wikipedia has written "A $k$-form is thought of as measuring the flux through an infinitesimal k-parallelepiped." How does a $k$-form do this? if this sentence is right, then the flux of which ...
0
votes
0answers
31 views
Finding a closed curve $\gamma$ on a Torus such that $\gamma$ is not pseudo-Anosov
Let g denote genus and n denote number of punctures. According to Kra's construction in Pseudo-Anosov theory for surfaces, if S is an orientable surface such that $3g+n>3$ and $\gamma \in ...
1
vote
0answers
23 views
Unique continuation for elliptic operators
Consider the following system of linear elliptic equations:
$ \Delta s_i = \sum_{j=1}^{d} l_{ij} s_j $ for $ i=1,\ldots d $ where $ l_{ij} = l_{ji} $.
It should be true that the following unique ...
0
votes
0answers
65 views
Formula for Gaussian curvature in terms of unit tangent vector fields?
Let $X\in\mathbb{R}^3$ be a surface with a local geodesic polar parametrisation with first fundamental form $du^2+G(u,v)dv^2$. How do we define unit tangent vector fields $e_1$, $e_2$ on $X$, forming ...
1
vote
0answers
67 views
Difference between parallel transport and derivative of the exponential map
Given a Riemannian manifold $M$, let $c(t) = \exp_p(tX)$ be the geodesic emanating from $p \in M$ with initial value $X$. Let $t_0$ be small enough, then we have to ways to map $T_pM$ to $T_{c(t_0)} ...
3
votes
1answer
82 views
Zeros of the second fundamental form
Let $ f:M \rightarrow N $ be a minimal immersion (of arbitrary codimension or an hypersurface if it is necessary) and let $ |A| $ be the norm of its second fundametal form.If $ A $ is not identically ...
4
votes
1answer
62 views
Elliptic equation on riemannian manifolds
Let $ M $ be a compact Riemannian manifold with or without boundary) and let $ \Delta $ be the metric laplacian. I want to study the differential operator $ -\Delta +q $ where $ q $ is a smooth ...
0
votes
0answers
45 views
Strongly parabolic PDE vs weakly parabolic PDE
In my studies on the Ricci flow, I was faced with a problem.
To prove the existence and uniqueness of solutions to the Ricci flow, it is proved that the Ricci flow is a Parabolic PDE type. Then one ...
4
votes
1answer
107 views
Why do we need Lie derivative?
If a manifold is equipped with Levi-Civita connection, Why do we need Lie derivative?
In Euclidean space to calculate directional derivative of a vector field V along W, we parallel transport V along ...
5
votes
1answer
70 views
A question about concept of pushforward
In An Introduction to Smooth manifolds by Lee is written: for any smooth vector fields V and W on a manifold $M$, let $\theta$ be the flow of $V$, and define a vector $(\mathcal{L}_v W)_p$ at each ...
4
votes
0answers
96 views
geometric meaning of Ricci-flatness
What is the geometric meaning of Ricci-flatness?
We know that if the Riemann tensor at a point vanished, manifold is flat at this point. but I don't know When the Ricci tensor vanished at a point, ...
2
votes
0answers
65 views
A question from Hamilton's Ricci Flow book by bennett chow
On page 3 of the book before exercise 1.2, has written: "torsion free is a compatibility condition with the differentiable structure".
I correctly do not understand how torsion-free condition results ...
1
vote
1answer
57 views
Compact surfaces without conjugate points
I've asked this question (Surfaces without conjugate points) and received an attentive answer from user67582. The answer made me see that I should ask better. So i'm trying again here.
I'm trying to ...
2
votes
1answer
45 views
Surfaces without conjugate points
I'm trying to understand some aspects of the geodesics of surfaces without conjugate points and the following question came out: consider the hyperbolic space and two geodesics wich are asymptotic to ...
2
votes
2answers
113 views
Normal coordinates
Let $M$ a riemannian manifold and $\nabla$ the Levi-Civita conection. Ineed to prove the next.
Let $B$ an open ball of radius $r$ in $T_pM$ such that $exp_p\mid _B$ be a difeomorphism over an open $U$ ...
3
votes
1answer
61 views
The intuition behind the definition of geodesics on a Riemannian manifold. (A non-technical question)
In the text I'm studying, the idea behind the definition of a geodesic on a Riemannian manifold was sketched via paths in $\mathbb{R}^n$. I have trouble understanding some aspects of it.
Let $\gamma: ...
